1. Limited time only! Sign up for a free 30min personal tutor trial with Chegg Tutors
    Dismiss Notice
Dismiss Notice
Join Physics Forums Today!
The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

Homework Help: Variant of the Maximum-Modulus Principle

  1. Nov 11, 2009 #1
    1. The problem statement, all variables and given/known data
    Show that the maximum principle holds with [tex]\phi :\mathbb{C}\rightarrow\mathbb{R}[/tex], 

    [tex]\phi(z) = (\textrm{Re}(z))^4 + (\textrm{Im}(z))^4[/tex],

    in place of the modulus:

    If [tex]U \subset \mathbb{C}[/tex] is open and connected, [tex]f : U \rightarrow \mathbb{C}[/tex] is holomorphic and [tex]p \in U[/tex] is such that
    
    [tex]\phi(f(p)) \geq \phi(f(z)) [/tex]

    then [tex]f[/tex] is constant.

    Can one also replace the modulus by the sine of the modulus, i.e. the function with [tex]\phi(z) = sin(|z|)[/tex]?


    2. Relevant equations
    If [tex]z=x+iy[/tex]

    [tex]\phi(z)=x^4+y^4[/tex]


    3. The attempt at a solution
    I tried using the open mapping theorem and the fact that

    [tex]f(U)\subseteq\{z|\phi(z)\leq\phi(f(p))\}=\{z|x^4+y^4\leq M\}[/tex]

    where [tex]M=\phi(f(p))[/tex], but was not successful.
     
  2. jcsd
Share this great discussion with others via Reddit, Google+, Twitter, or Facebook

Can you offer guidance or do you also need help?
Draft saved Draft deleted